Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.
The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.
A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.
Primitive translation vectors are used to define a crystal translation vector, and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors, and are primitive if the atoms look the same from any lattice points using integers, and .
The primitive cell is defined by the primitive axes (vectors), and . The volume, of the primitive cell is given by the parallelepiped from the above axes as,
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—Blaise Pascal (1623–1662)